Blown head gasket?

Had issues with heat not working and then it got hot one time so I changed thermostat and may have not fully bled air out but I tried using the bleeder valve and it shoots out a stream when the cars off but seems to get bubbles out of it once the car starts. I tried a leak down test and the fluid seems to stay blue like there is no exhaust leaking into the coolant, and I definitely cannot get any substantial heat from it either. I am wondering if a bad water pump could possibly be making it overheat and the temp sensor does not catch it because its not circulating. The t-stat does work I put it in boiling water, and also one hose gets hot while the other stays cool. I thought steady bubbles in the no-spill funnel after an hr or so was a sure sign of a blown head gasket, is that not the case? Also is it possible to have a false positive on the block tester? I have the t-stat back out now and it seems like there is no blockage in the system b/c it moves freely, also I ran a compression test on the coolant system and it passed that as well.